Overexpression is increasing the amount of a particular protein in a cell line or organism. It is usually done either be introducing an expression plasmid into cells or by introducing exogenously-transcribed mRNA into cells. This is a common technique in embryo studies, where (at least for zebrafsih, Xenopus, sea urchins and Ciona) the plasmid or mRNA us generally microinjected into the early zygote though a thin glass needle so that after mitosis the daughter cells contain the new nucleic acids and overexpression is maintained though some of the development process.

A rescue, at least in the context I'm familiar with them, is when an endogenous gene is knocked down and an exogenous nucleic acid is introduced to compensate for the lack of protein production from the endogenous gene. This is commonly done in gene knockdown experiments to show that the observed phenotypic change caused by the knockdown is due to suppression of the target mRNA and not due to interaction of the knockdown agent (commonly an oligo) with a different, unexpected mRNA.

Here's a narrative example. You want to know what gene A does during development. You microinject an oligo to knock down gene A and you get peculiar embryos, perhaps with small heads, eyes migrated forward and fused and with a single shared lens (anyone recognize that gene?). You then microinject both an oligo to knock down gene A and a modified mRNA for gene A that is missing the oligo target sequence. This time, the embryos develop normally -- you've "rescued" the knockdown phenotype. This outcome shows that, since adding more expression of gene A relieved the phenotype, the observed phenotype was due to knockdown of gene A and not accidental knockdown of gene B due to off-target interactions fo the knockdown reagent.

Knockdowns can't always be rescued in developmental systems. Sometimes the time or place of expression of a gene is critical to proper development. When you microinject a rescue mRNA, you get translation in all cells at all times until the mRNA is degraded. That expression at the wrong place and time might not successfully rescue a knockdown phenotype -- you could mess up the embryo's development even more. The rescue is a beautiful control when it works, but can be an expensive wasted experiment of you are trying to rescue a gene that can't be rescued due to developmental defects induced by ectopic expression.

thanks @jonmoulton!So the technique of introducing something for over-expression or a genomic rescue is the same? For example in a fly, you can use GAL4-UAS system for over-expression. You can do the same thing for a rescue experiment but just in a null background?

That's it. Whether you are doing overexpression or a rescue depends on whether the endogenous gene is also expressing. Though if you add too much rescue RNA, you can end up overexpressing as well as rescuing.